Regularizers for Vector-Valued Data and Labeling Problems in Image Processing

Regularizers for Vector-Valued Data and Labeling Problems in Image Processing

Дан обзор последних результатов в области регуляризаторов, основанных на полных вариациях, применительно к векторным данным. Результаты оказались полезными для хранения или улучшения мультимодальных данных и задач разметки на непрерывной области определения. Возможные регуляризаторы и их свойства ра...

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Date:2011
Main Authors: Lellmann, J., Schnörr, C.
Format: Article
Language:English
Published: Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України 2011
Series:Управляющие системы и машины
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Оптимизационные задачи структурного распознавания образов
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/82923
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Cite this:Regularizers for Vector-Valued Data and Labeling Problems in Image Processing / J. Lellmann, C. Schnörr // Управляющие системы и машины. — 2011. — № 2. — С. 43-54. — Бібліогр.: 50 назв. — англ.

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spelling nasplib_isofts_kiev_ua-123456789-829232025-02-10T01:19:20Z Regularizers for Vector-Valued Data and Labeling Problems in Image Processing Регуляризаторы векторных данных и задачи разметки в обработке изображений Регуляризатори векторних даних та задачі розмітки в обробці зображень Lellmann, J. Schnörr, C. Оптимизационные задачи структурного распознавания образов Дан обзор последних результатов в области регуляризаторов, основанных на полных вариациях, применительно к векторным данным. Результаты оказались полезными для хранения или улучшения мультимодальных данных и задач разметки на непрерывной области определения. Возможные регуляризаторы и их свойства рассматриваются в рамках единой модели. The review of recent developments on total variation-based regularizers is given with the emphasis on vector-valued data. These have been proven to be useful for restoring or enhancing data with multiple channels, and find particular use in relaxation techniques for labeling problems on continuous domains. The possible regularizers and their properties are considered in a unified framework. Наведено огляд останніх результатів у галузі регуляризаторів, що базуються на повних варіаціях, стосовно векторних даних. Результати виявилися корисними для зберігання та покращення мультимодальних даних і задач розмітки на неперервній області визначення. Можливі регуляризатори та їх властивості розглядаються в рамках єдиної моделі. 2011 Article Regularizers for Vector-Valued Data and Labeling Problems in Image Processing / J. Lellmann, C. Schnörr // Управляющие системы и машины. — 2011. — № 2. — С. 43-54. — Бібліогр.: 50 назв. — англ. 0130-5395 https://nasplib.isofts.kiev.ua/handle/123456789/82923 004.93’1:519.157 en Управляющие системы и машины application/pdf Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic Оптимизационные задачи структурного распознавания образов
Оптимизационные задачи структурного распознавания образов
spellingShingle Оптимизационные задачи структурного распознавания образов
Оптимизационные задачи структурного распознавания образов
Lellmann, J.
Schnörr, C.
Regularizers for Vector-Valued Data and Labeling Problems in Image Processing
Управляющие системы и машины
description Дан обзор последних результатов в области регуляризаторов, основанных на полных вариациях, применительно к векторным данным. Результаты оказались полезными для хранения или улучшения мультимодальных данных и задач разметки на непрерывной области определения. Возможные регуляризаторы и их свойства рассматриваются в рамках единой модели.
format Article
author Lellmann, J.
Schnörr, C.
author_facet Lellmann, J.
Schnörr, C.
author_sort Lellmann, J.
title Regularizers for Vector-Valued Data and Labeling Problems in Image Processing
title_short Regularizers for Vector-Valued Data and Labeling Problems in Image Processing
title_full Regularizers for Vector-Valued Data and Labeling Problems in Image Processing
title_fullStr Regularizers for Vector-Valued Data and Labeling Problems in Image Processing
title_full_unstemmed Regularizers for Vector-Valued Data and Labeling Problems in Image Processing
title_sort regularizers for vector-valued data and labeling problems in image processing
publisher Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України
publishDate 2011
topic_facet Оптимизационные задачи структурного распознавания образов
url https://nasplib.isofts.kiev.ua/handle/123456789/82923
citation_txt Regularizers for Vector-Valued Data and Labeling Problems in Image Processing / J. Lellmann, C. Schnörr // Управляющие системы и машины. — 2011. — № 2. — С. 43-54. — Бібліогр.: 50 назв. — англ.
series Управляющие системы и машины
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fulltext УСиМ, 2011, № 2 43 UDC 004.93’1:519.157 J. Lellmann, C. Schnörr Regularizers for Vector-Valued Data and Labeling Problems in Image Processing Дан обзор последних результатов в области регуляризаторов, основанных на полных вариациях, применительно к векторным данным. Результаты оказались полезными для хранения или улучшения мультимодальных данных и задач разметки на непре- рывной области определения. Возможные регуляризаторы и их свойства рассматриваются в рамках единой модели. The review of recent developments on total variation-based regularizers is given with the emphasis on vector-valued data. These have been proven to be useful for restoring or enhancing data with multiple channels, and find particular use in relaxation techniques for la- beling problems on continuous domains. The possible regularizers and their properties are considered in a unified framework. Наведено огляд останніх результатів у галузі регуляризаторів, що базуються на повних варіаціях, стосовно векторних даних. Результати виявилися корисними для зберігання та покращення мультимодальних даних і задач розмітки на неперервній об- ласті визначення. Можливі регуляризатори та їх властивості розглядаються в рамках єдиної моделі. Abstract We review recent developments on total varia- tion-based regularizers, with emphasis on vector- valued data. These have been proven to be useful for restoring or enhancing data with multiple chan- nels, and find particular use in recent advances on relaxation techniques for multiclass labeling prob- lems on continuous domains. Many of the propo- sed approaches only differ in the norm that is used. We provide a review of the possible regular- izers and their properties in a unified framework. 1. Introduction Total variation-based variational approaches in an image processing exhibit some unique proper- ties. Most prominently, they can be viewed as a number of «stacked» problems on the sublevelsets of a function, which allows to gain insight into the structure of their minimizers, and can be applied to finding global minimizers of segmentation/labe- ling problems formulated on continuous domains. This work is intended as an introductory over- view of the field. We first discuss the related func- tionals that are commonly used in image process- ing (Sect. 2). Although they were originally for- mulated for scalar-valued total variation regulariz- ers, they can be readily extended to vector-valued data. This occurs in particular when dealing with convex relaxations of segmentation problems. Recently, many different total variation-based regularizers have been proposed in different con- texts. We try to give a systematic account of the Keywords: variational image segmentation, image label- ing, convex relaxation, nonsmooth convex optimization. existing approaches and their properties (Sect. 3). Although the intention of this work is to provide a broad overview on the subject, some technicalities cannot be avoided and will be defined in the fol- lowing section. Preliminaries In the following, superscripts vi denote a collec- tion of vectors or matrix columns, while sub- scripts vk denote vector components or matrix rows, i.e. we denote, for d lA  , .)||(=)||(= 1 1  d l aaaaA  (1) We denote by := { | 0, = 1}l l x x e x   the unit simplex in l , where := (1, ,1) le    . The i-th unit vector is denoted by ie , nI is the identity matrix in n and 2  the usual Euclid- ean norm for vectors resp. the Frobenius norm for matrices. Analogously, the standard inner product , extends to pairs of matrices as the sum over their element-wise product. )(xr denotes the ball of radius r in x, and 1dS the set of dx with x =1. The indicator function )(1 x of a set  is defined as 1=)(1 x iff Sx and 0=)(1 x oth- erwise. For a convex set  ,   vuu v ,sup:=)(  is the support function from convex analysis. For simplicity, we assume that the the image domain d  is the open unit box, d(0,1)= . All results can equally be formulated for bounded open domains with compact Lipschitz boundary. The images or generally data with a spatial do- main  are represented as vector-valued func- 44 УСиМ, 2011, № 2 tions : lu  which are absolutely integrable, i.e. lLu )(1  . )(k cC is the space of k-times continuously differentiable functions on  with compact sup- port. As usual, d denotes the d-dimensional Le- besgue measure, while k denotes the k-dimen- sional Hausdorff measure. We say that : lu  is of bounded varia- tion, i.e. lBVu )( , if lLu )(1  and its distribu- tional derivative corresponds to a finite Radon measure; i.e. )(1  Lu j and there exist d -valued measures ),,(= 1 jdjj uDuDDu  on the Borel subsets )( of  such that =1 =1 =1 div = ( ( )) . l l d i j j j i j j j i d l c u v dx v dD u v C            (2) The measures in (2) form the distributional gradient Du , which is again a measure that van- ishes on any 1)( d -negligible set. The total variation of u is defined as the meas- ure-theoretic total variation of its distributional gradient [AFP00, Prop. 3.6]: For some (possibly vector-valued) measure  on the measure space ),( X , the total variation measure of  on a set E is defined as 2 =0 | | ( ) := sup ( ) ( )k k k E E E        pairwise disjoint, =0 = .k k E E      (3) Consequently, the total variation of lLu )(1  is defined as .||1=)(|:=|)( DudDuuTV  (4) The total variation also has the dual representa- tion ( ) , ( ) 1 ( ) = , Div ,sup d lv C v xc TV u u v dx       (5)  1Div := div , ,div . Τ lv v v By partial integration, this implies that for con- tinuously differentiable lCu )(1  , 2( ) = ,TV u u dx   (6) however the definition of the total variation allows u to be discontinuous or even piecewise constant. For more general functionals depending on Du , first note that Du can be represented in terms of its density with respect to || Du , i.e. Du = |||)|/(= DuDuDu . Since ||/ DuDu is actually a || Du -integrable function, the total variation is 2 ( ) = | | . | | Du TV u d Du Du (7) Based on this principle it is possible to con- struct non-standard TV-based regularizers: For 0: d l    continuous, positively homogene- ous and convex, we define the measure .|| || :=)( Du Du Du Du        (8) Essentially  generates a new measure from Du by transforming its density with respect to || Du [AFP00, Thm. 2.38]. Note that by definition  is a seminorm, and a norm if ( )z 0=0= z . 2. TV-Based Functionals in Image Analysis In this paper, we will be concerned with a par- ticular class of variational problems used in image processing and analysis. We refer to [SGG+09] for a general overview. The output of a variational method is defined as the minimizer ),(minarg:= ufu u   (9) where  is some subset of a space of functions defined on  , and f a functional depending on the input data. The interpretation of u is governed by the problem to be solved: for the prototypical ex- ample of color denoising, 3[0,1]: u could directly describe the colors of the output image on the image domain d  ; while for segmenta- tion problems, [0,1]: u could assign each point to the foreground ( 1=)(xu ) or background ( 0=)(xu ) class. We will in particular consider the case where u is vector-valued. Usually the objective f is composed of a data term )(uH and a regularizer )(uJ , ).()(=)( uJuHuf  (10) УСиМ, 2011, № 2 45 The data term strongly depends on the input da- ta – such as color values of a recorded image, depth measurements, or other features – and pro- motes a good fit of the minimizer to the input data. However, due to the presence of noise in the input data, it is generally necessary to incorporate the additional prior knowledge about the «typical» appearance of the desired output, which is the pur- pose of the regularizer. We will now consider some important choices for data term and regularizer. 2.1. L2 – L2: Gaussian Denoising The central question is how to construct the in- dividual terms, and which combinations lead to go- od results. The most basic, classical example is Gaussian denoising, where the goal is to find, given an input image : lI  , some : lu  mi- nimizing 2 2 2 2 1 ( ) = 2 2 f u u I dx Du dx       (11) for some weighting parameter  >0. Usually u is re- quired to be differentiable, nevertheless we use the distributive derivative Du to simplify comparison with the following variants. Note that both the da- ta term and the regularizer exhibit quadratic growth. The problem is convex, and after the discretiza- tion can be solved globally optimal as a linear equa- tion system. While the approach removes Gaussian noise very well, it tends to smear hard edges in the image. This is caused by the quadratic growth of the regularizers, which makes it susceptible to «outli- ers» – i.e. high gradients – in the form of hard edges. 2.2. L2 – TV: Rudin-Osher-Fatemi Many approaches have been proposed to cir- cumvent this problem. Most prominently, the ani- sotropic diffusion approach consists in solving (11) using gradient descent, at each step locally modifying the norm in the regularizer to reduce smoothing across directions where the current it- erate has a large gradient, i.e. across potential edges. While this is widely used and gives good results in many cases, the output cannot be charac- terized in the variational way as the minimizer of a certain functional. A more one-step approach is Rudin-Osher-Fatemi ( ROF ) denoising [ROF92], 2 2 2 1 ( ) = 2 f u u I dx Du       . (12) The right-hand side should be seen as a short- hand notation for the total variation of u. The sca- lar-valued case has been extensively studied and works well for removing Gaussian noise while preserving hard edges; also, the overall problem is still convex and therefore can be solved globally optimal. The key difference is that while the data term still has quadratic growth, the regularizer only grows linearly. In this paper, we will be concerned with regularizers that also have this property. 2.3. L1 – TV For non-Gaussian noise such as salt-and-pep- per, (12) is suboptimal, as it is quite sensitive to outliers in the input image I. Also, ROF denoising invariably leads to a reduction in contrast. These drawbacks are addressed by the L1 – TV model (see [Nik01] for an overview), 1 2( ) = .f u u I dx Du dx       (13) Here, both the data term as well as the regular- izer exhibit linear growth. Existence and well- posedness of (12) and (13) can be shown in a pre- cise sense within the class of functions of bounded variation [AFP00, AMT91]. The functionals such as (13) are extremely tol- erant to noise. The downside is that they also tend to generate a «staircasing» effect on smooth gra- dients, i.e. the solution tends to be piecewise con- stant. We refer to [DAG09] and the references therein for a detailed analysis. For image denois- ing this is certainly not desirable, therefore some effort has been put into reducing staircasing while preserving robustness, mostly by including the higher-order derivatives (see e.g. [CL97, Sch98, CMM00, LT06, BKP10]). It should also be noted that the robustness can be increased even more at the cost of convexity [Nik]. 2.4. Linear– TV: Scalar Case Upon closer inspection there are applications where staircasing – or more specifically piecewise constancy – is exactly what is desired for the out- put. This was noted in [CEN06] for the case of geo- metry denoising: Here the goal is to find, for some given set T , a set S that minimizes ),(|:=|)( SPerSTSf  (14) 46 УСиМ, 2011, № 2 where |)\()\(|:=| | TSSTST  is the volume of the symmetric difference of the sets T and S, and Per(S) is the perimeter of S, Per(S) := TV(1S), which coincides with the classical length resp. area of the boundary if S is sufficiently smooth. Setting Su 1:= and relaxing )(xu to the whole of R, we arrive at the problem of minimizing 2( ) = | 1 | ,Tf u u dx Du      (15) over )(BVu , which is a TVL 1 denoising pro- blem as in (13) with TI 1= . The important aspect is that we departed from the requirement that u must be an indicator function, allowing intermedi- ate values. This leads to an overall convex prob- lem (15) which can be solved globally optimal; however it introduces the possibility that we may obtain non-discrete solutions, which do not corre- spond to an indicator function. In this case it is very desirable that the mini- mizer is piecewise constant. Ideally it should be a discrete solution, i.e. take only the values 0 or 1. In fact this is often observed in practice. More- over, in [CEN06] it was noted that any non- discrete solution may be turned into a discrete one by a simple thresholding (i.e. rounding) of u. This allows to solve combinatorial problems globally optimal by solving convex problems and post- processing the solutions obtained. This thresholding property applies to the much wider class «Linear-TV» problems 2( ) = ( ) ( ) ,f u u x s x dx Du     (16) with the constraint u  BV(). As shown in [CEN06], this allows to solve the Chan-Vese two-class im- age segmentation problem [CV01]. Here the de- sired image is piecewise constant, taking only one of two grey values, 0c or 1c . Each point in the im- age should be assigned either to the foreground ( 1=)(xu ) or the background ( 0=)(xu ), such that     2 1 2 0 2 ( ) = ( ) (1 ) ( ) f u u c I x dx u c I x dx Du             (17) is minimized over all indicator functions u  BV(). For fixed 1 2,c c  , this is equivalent to a problem of the form (16) for s = (c1 – I (x))2 – (c0 – I (x))2. In a sense, the problem (16) extends the con- cept of graph cut-based segmentation – formu- lated on a finite set of points in the image domain, typically a uniform grid – to continuous image domains. Therefore it is also referred to as a con- tinuous cut . For an extensive analysis of a closely related problem and its associated dual problem we refer to [Str83]. An important property is that solutions of the ROF and L1 – TV problems (12), (13) for scalar- valued u are intimately connected with solutions of associated continuous cut problems of the form (16). This comes from the fact that the scalar total variation can be written in terms of it level sets, .)(1=)( }>)(|{    dTVuTV xux (18) This coarea formula [FR60] intuitively con- nects the problem of finding Ru : with a se- ries of problems of finding indicator functions for problems of the type (16), and can be exploited in two directions: On the one hand, it allows to solve a family of parametrized continuous cut problems by solving a single ROF-type problem and thresh- olding [CD09, Ber09, SKO09]. On the other hand, ROF- and L1 – TV type problems can be solved by solving a series of continuous cuts to find the su- per-level sets }>)(|{  xux for all  , and from these reconstructing u [Hoc01, DS06a, DS06b, GY07, CD09, DAG09]. While in theory this would require to solve in- finitely many problems, in practice the range of va- lues of u is often quantized, such as {0, 1, …, 256} for greyscale images. Under a suitable discretiza- tion that respects the coarea formula [CD09], it is possible to apply the same results to find the quan- tized u – originally a combinatorial problem – by solving a finite number of continuous cuts, or even graph cuts [DS06a, DS06b, DAG09]. 2.5. Linear TV : Multiclass Labeling The results from the previous section do not transfer to the case of vector-valued u, as there is no natural extension of the coarea formula (18) and of the thresholding process to more than two classes. However it is still interesting to consider the extension of (16) to vector-valued u, 2( ) = ( ), ( ) .f u u x s x dx Du       (19) УСиМ, 2011, № 2 47 This problem class naturally occurs in relaxa- tions of multi-class image labeling problems . Here the goal is to find, for each pixel x in the im- age domain d  , a label },{1,)( lx   which assigns one of l class labels to x so that the labe- ling function  adheres to some data fidelity and spatial coherency constraints. In contrast to the ima- ge restoration applications above, u must assume one of a finite number of values at each point. Several authors [ZGFN08, CCP08, LKY+09] independently proposed linearization techniques for this problem; the basic idea is also equivalent to the basis-function technique in [LLT06], and is a continuous counterpart to solving combinatorial finite-dimensional problems using LP relaxation [KT99]. We follow the notation used in [ZGFN08, LKY+ +09]: Identify label i with the i-th unit vector i le  , set },,{:= 1 leeE  , and find Eu : minimizing 2( ) = ( ), ( ) .f u u x s x dx Du       (20) The data term assigns to each label u(x) = e i a local cost )(xsi . If one constrains lBVu )( , the right-hand side is well-defined even though u is not necessarily differentiable, and represents the total weighted length of interfaces between re- gions of constant labeling. As the data term is lin- ear, the local costs s may be arbitrarily complex without affecting the overall problem class. To tackle the combinatorial nature of (20), the problem is relaxed,  2( ), ( ) ,inf u u x s x dx Du        (21) where 1}=)(0,)(|)({:= 1= xuxuBVu i l ii l   is a convex set which constrains each )(xu to the unit simplex, i.e. to the convex hull of E. Since the regularizer is still convex, the overall problem is as well. On the other hand, due to the relaxation artificial non-discrete solutions with u(x)  E at some points may be introduced. As the data term was linearized, the local costs s may be arbitrarily complex, possibly derived from a probabilistic model, without losing con- vexity. Thus the important question is what effects can be achieved by modifying the regularizer. In the scope of this paper, we will consider ap- proaches where the regularizer is replaced by ( ) := ( ),J u Du   (22) 0where : l d    is continuous, convex, po- sitively homogeneous, i.e. )(=)( zccz  for 0>c , and satisfies u z 2 ( ) lz z    for some 0>lu   . In the TVLinear  case, exis- tence of a solution for the problem  ,)()(),(inf Dudxxsxu u    (23) follows from [AFP00, LS10]. Although we mainly motivated the use of mo- dified regularizer in the labeling framework, they are as well interesting for ROF- and 1LTV  type problems (12), (13) applied to vector-valued data. 3. Total Variation-Based Regularizers We will first review some basic properties of the custom regularizer (22). For the proofs and details we refer to [LS10]. 3.1. Basic Properties A structure of the regularizer. Consider the effect of the regularizer (22) on labeling functions u that takes only two discrete values, ie and je for some },{1,, lji  , i.e. S j S i eeu \11=  for some S with <)(SPer . Then by [AFP00, Thm. 3.84] (a variant of this was also shown in [LS10, Prop. 4]),    1 \( 1 1 ) = ,i j i j d S S SS J e e e e d          (24) where S denotes the reduced boundary of S, and 1:  d S SS is the generalized inner normal of S [AFP00, Def. 3.54]. Therefore the regularizer locally penalizes jumps along the (reduced) boundary of S depend- ing on the normal at each point, and on the labels of the adjoining regions. Isotropy Often, the regularizer is assumed to be rotation invariant ( isotropic ): )()(=)( dSORzRz  (25) This is obviously the case if and only if )(=)( 1   xex for any 1 dS . If  is iso- 48 УСиМ, 2011, № 2 tropic, for the application of multiclass labeling we may define the interaction potential   .:=),( 1   ji eeejid (26) Due to the isotropy, we have  =)(1  ji eee  )((=))(= 11 ijji eeeeee   , therefore d(i, j) = d(j, i). From convexity and positive ho- mogeneity it follows that d must be subadditive, and from the lower bound l we get that d(i, j) = = 0 i = j. Hence for isotropic regularizers, the interaction potential d must be a metric, and defines the be- havior on label functions taking only discrete label values (e1, , el) in the spirit of (24) by \ 1 ( 1 1 ) = ( , ) = ( , )Per( ), i j S S d S J e e d i j d d i j S         (27) i.e. boundaries between regions of constant label- ing are penalized by their length, weighted by d(i, j) depending on the labels i, j of the adjoining regions. Permutation Invariance. This refers to the in- variance of the regularizer with respect to permu- tations of the elements of u, i.e. of the label set in multiclass labeling. In terms of , invariance is given if (z) = (zP) for any permutation matrix l lP  . Separability. If  can be written as a sum of terms that depend only on individual components or directional derivatives of u, it is called separa- ble in the components of u resp. in space. Separa- bility usually simplifies optimization, as it reduces the coupling between variables. Homogeneity. In (21) we assume spatial ho- mogeneity (translation invariance), as  does not depend on x. It is also possible to consider general non-homogeneous regularizers of the form ).(Dux (28) This is a common practice for anisotropic  in combination with a process where the anisotropy is controlled by local properties of the input or the current iterate. An even more general regulariza- tion approach is ),,,( Duuxg (29) however this considerably complicates the re- quirements for the existence of solutions [AFP00, Chap. 5], and is out of scope for this paper. Dual Formulation. In analogy to the dual defi- nition for the total variation, for any continuous, positively homogeneous and convex  it is pos- sible to give a dual definition: By Fenchel duality,  is the support function of some closed convex set loc d l  [RW04], .,sup=)(=)(   yzzz locyloc   (30) Then, in analogy to the dual formulation of the total variation, },|,{sup=)(  vdxDivvuuJ (31) }.)(|)({:=   xxvCv loc ld c  (32) Thus  can be defined implicitly by its dual set loc . This representation is also convenient for dual or primal-dual optimization methods [Cha05, ZGFN08, CCP08, LBS10, LS10]. 3.2. Isotropic Approaches 3.2.1. Frobenius Norm The most classical choice for  is the Froben- ius norm,   .:=)( 2 1 2 ,         j i ji F uDDu (33) This definition is the basis for large parts of geometric measure theory and the theory of func- tions of bounded variation [AFP00], and is some- times referred to as MTV in the context of denois- ing of vector-valued data [SR96, DAV08]. It is isotropic and permutation invariant, however it is neither separable in the components of u nor in space. The dual set F loc is just the 2 unit ball in d l . The associated potential is     1/ 2 = 1 =: ( , ).i j i j ue e d i j    (34) The potential ud on the right-hand side is known as uniform, discrete, or Potts metric, and widely used in approaches defined on finite grids [KT99, BVZ01, KT07]. Optimization approaches for more involved discretizations can be found in [LLT06, LKY+09]. УСиМ, 2011, № 2 49 3.2.2. Linear Transformations A straightforward modification that can be applied to most  is to introduce a matrix  1= | | l k lA a a   for some lk , and define   .=))((:=)(,  ADuAuDDu FFAF (35) This corresponds to substituting the Frobenius matrix norm on the distributional gradient with a linearly weighted variant. While we only consider the Frobenius norm here, the approach can in principle be used to augment all other norms. It retains isotropy of the underlying norm, but nei- ther permutation invariance nor separability. Applied to a jump from label i to label j, this results in the potential   , 2= =: ( , )i j i j F A Ae e a a d i j    .(36) As noted in [LBS09], metrics of the form Ad are known as Euclidean metrics. This class com- prises some important special cases:  the uniform metric, with IA )2(1/= ;  the linear (label) metric, ||=),( jicjid  , with ),,,2(= lcccA  . This regularizer is suitable to pro- blems where the labels can be naturally ordered, e.g. the depth from stereo or grayscale image de- noising.  More generally, if label i corresponds to a prototypical vector zi in k-dimensional feature space, and the Euclidean norm is an appropriate metric on the features, it is natural to set ( , ) = i jd i j z z , which is Euclidean by con- struction. This corresponds to a regularization in feature space, rather than in «label space». Also, non-Euclidean metrics such as the trun- cated linear metric, |}|{2,min=),( jijid  , can be approximated by solving a (convex) SDP prob- lem, cf. [LS10] and the references therein. One major advantage of this kind of modifica- tion is that it is quite powerful, but retains a clo- sed-form expression for the regularizer. The dual set is just A= F loc A loc  . Alternatively, it is often easier to formally merge A into the linear gradient operator Du, which allows to keep the structure of the dual set and requires only few modifications to the optimization method [LKY+09]. 3.2.3. Channel-By-Channel Maybe the most straightforward approach to transfer from scalar total variation to the vector- valued case is to sum up the total variations of the components, i.e. 1 1 2 2( ) := .lDu Du Du   (37) In this formulation, the objective is separable in the components of u, which potentially simplifies numerical optimization [Blo98, ZGFN08]. For unconstrained u, a variant of the coarea formula (18) still holds, i.e.    1 1 >( ) = ( ) , :=1 .ui i Du Dw d w            (38) However for labeling problems, where one is looking for 1: = { , , }lu E e e  , this is of very limited interest, since Exw  )( in general. Similar to the Frobenius norm, 1 implements the uniform metric,     11/ 2 = ( , ),i j ue e d i j   (39) and is isotropic, with 2 11 |)||{(=  il loc zzz  }},{1,1 li  . As in the case of the Frobenius norm, linearly transformed variants of the form     AuDDuA 11, :=  (40) could be used. A straightforward transformation shows that 1, 1= i j A a a  . The class of metrics covered by this approach would thus again be those representable using a linear embedding into a space that is now endowed with the norm given by 1 instead of the Euclidean norm. 3.2.4. Convex Envelope Approach Motivated by [ABDM01], Chambolle et al. [CCP08] proposed an approach for constructing regularizers for multiclass labeling problems, where the potential d is given in advance as |)(|=),( jijid  for a positive concave function  . While they used a different notation for the pa- rametrization of the unit simplex through u, their parametrization is equivalent to (20) under a linear transformation of the components of u. The approach is derived by setting          ., ,)(=,|| :=)( otherwise eezji z ji (41) 50 УСиМ, 2011, № 2 Then  is constructed as the convex envelope of  , i.e. the largest convex function smaller or equal to  . This potentially generates  that are as large as possible while still satisfying (27) for the given d, and thus we may hope that the relaxa- tion to the unit simplex (21) does not generate too many artificial non-discrete solutions. This leads to   loc  1:= { = , , l i j v v v   2 ( ), = 0}.d l i jv v i j e v       (42) The approach can be extended to arbitrary met- rics d by setting [LS10] d loc  1:= { = , , l i j v v v   2 ( , ), = 0}d l i jv v d i j e v      (43) for some given interface potential ),( jid . By definition, d loc and thus d loc d   = are isotropic. In [LS10] it was shown that, for metrics d, the re- sulting d satisfies the desired ).,(=))(( jidee ji d  (44) This provides an approach to construct regular- izers with arbitrary (metric) prescribed interaction potentials d. The downside is that there is no sim- ple closed expression for  and thus for the regu- larizer, and the dual set can be quite involved, which potentially complicates optimization. From (44) it can be seen that d is permuta- tion invariant only for 0,=  udd . Also note that in order to define d, d does not have to be a metric. However (44) then only holds as an ine- quality, so J is not a true extension of the desired regularizer. 3.2.5. Eigenvalue-Based Norms In the anisotropic diffusion community it is widely used practice ([SR96], see also [WS01] and the references therein) to employ weighted norms based on the eigenvalues 01  d  of the structure tensor    .=)( DuDuDuG (45) Let ),,(:=)( 22 1 2 dDu   , i.e. the i represent the magnitudes of the singular values of Du. While originally rooted in a diffusion framework, the approach can also be used to construct TV-like regularizers. It includes the Frobenius norm, since .)(=)( 2 DueDuF   (46) In addition, for some rotation matrix ddRR  and permutation matrix llRP  ,    ,)( =)(=)((    RDuRG RDuPPDuRPDuRG (47) therefore  2 2( ) = ( )Du R Du P  , i.e. all norms derived from these singular values are isotropic and permutation invariant. The Frobenius norm approach was considered for color denoising in [Blo98]. They observed that, since 2 1/2( ) := ,F i i Du Du        (48) the Frobenius norm prefers transitions with similar magnitude in all channels: The transition (0,0) (1,1) is assigned a much lower penalty than the two consecutive transitions (0,0)(1,0)(0,1). This phenomenon does not occur with the Chan- nel-by-Channel regularizer 1. For color denoi- sing, this leads to a color smearing effect at edges and a color shift towards the greyscale image [Blo98]. A similar effect was observed in [CCP08] for multiclass segmentation, where the preference towards similar gradients leads to minimizers that assume non-discrete values more frequently than is the case for e.g. d . In [GC10a] it was proposed to employ ,)(:=)( 2 12 GDu  (49) which amounts to the standard 2 operator norm on Du. The corresponding dual set can be repre- sented as 2 2 2= { | , , 1, 1}.d l loc x R x R x       (50) While this more difficult to handle numerically than e.g. F loc , it can be dealt with reasonably well in primal-dual methods, and experimentally re- duces color smearing and channel coupling in de- noising, deblurring and superresolution applica- tions [GC10a]. Regarding labeling approaches, we have     21/ 2 = ( , ),i j ue e d i j   (51) УСиМ, 2011, № 2 51 i.e. 2 represents the uniform metric. However, since F 2 , for multiclass labeling the energy when using 2 potentially generates more unde- sired minima than the standard choice F . 3.3. Anisotropic Approaches In this section we will consider approaches that are not rotation invariant. Note that most of these have been developed for scalar-valued total varia- tion; however they extend to the vector-valued case in a straightforward manner and could be coupled with any of the above approaches for weighting different labels. 3.3.1. Wulff Shapes For scalar-valued u, the use of anisotropic vari- ants of the total variation has been studied in [EO04] for the ROF model, where the authors characterize minimizers of such functionals. They base their analysis on the «Wulff shape» associ- ated with  , which is identical to loc for the sca- lar-valued case. As an example, consider setting db loc [0,1]:= . Then, for (scalar!) )(BVu , 1 1( ) = =| | | | .d b Du Du D u D u   (52) Essentially, the Wulff shape defines the norm  via the unit ball of its dual norm. It can be shown that the structure of loc is reflected in the structure of the minimizer of the ROF functional (12) in the sense that it does not affect structures in the shape of loc itself: For  small enough, the minimizer u u of   21 ( ) = 1 ( ) 2 loc f u u Du       (53) is just a multiple of the input, i.e. loc cu 1= [EO04, Thm. 4.1]. Applied to the above definition for b , this means that the unit box may occur as the minimizer of the anisotropic ROF model, which cannot happen for the standard ROF model [Mey01]. Based on these results, it was shown in [ZNF09] that the thresholding property for isotropic con- tinuous cuts can be transferred to their anisotropic counterparts, i.e. it is still possible to recover dis- crete solutions of the anisotropic continuous cut problems by thresholding. When combined with a spatially varying, edge-driven adaptation of the Wulff shape, they observed improved visual qual- ity when applied to the reconstruction of depth maps and 3D structure. These anisotropies can also be extended to the vector-valued case, e.g. by setting ,)(=)( 2         ib i DuDu (54) or by replacing 2 by b in (43). However as in the isotropic case this invalidates the thresholding property. 3.3.2. Anisotropy from Discretization: 4-neighborhoods A large class of anisotropies that occurs in practice are actually induced by the discretization used to approximate the total variation on grids. A very common scheme is to add the total variation of the individual components, 1 , ( ) := .d a Du D u D u   (55) Here  refers to some norm on l , notable ca- ses include 2 and the completely separable case .||= :=)(:=)( 1=1= 1 1 1 1,,1 j i l j d i d aa uDuD uDDuDu       (56) This is in fact the anisotropy that is implicitly assumed by many algorithms that use a grid-based representation with pairwise potentials [KT99, BVZ01, KT07]: Assume that  = (0,1)2, N  N and h := 1/N, and u is discretized by its values on the uniform grid {xi,j = (ih, jh) i, j  {0, , N}}. The usual 4-neighborhood discretization then amounts to (with appropriate boundary conditions) approximating the directional derivatives via           1 , 1, , 2 , , 1 , 1 ( ) , 1 ( ) . i j i j i j i j i j i j D u x u x u x h D u x u x u x h       (57) In the case of scalar-valued u (or for the indi- vidual terms in ,1a ), the total variation with Neumann boundary conditions is then discretized as     1 1, , , =0 1 ( ) N i j i j i j J u u x u x h     52 УСиМ, 2011, № 2 +     1 , 1 , , =0 1 . N i j i j i j u x u x h    (58) This type of energy is tremendously popular as it is convex, contains only pairwise terms (i.e. terms depending on only two different variables) and is therefore easy to implement and analyze. Moreover, for two-class problems (resp. scalar- valued u ), the energy has the thresholding prop- erty and is submodular, which allows to find a discrete solution in polynomial time, for example using graph-cut methods [BVZ01]. For infinitesimal grid size, it implements the ,1a norm. The main drawback is that edges par- allel to the coordinate axes are preferred to diago- nal edges, which often leads to «zig-zag» artifacts on diagonal structures. 3.3.3. The anisotropy from Discretization: Larger Neighborhoods To some extent, the discretization-induced ani- sotropy can be reduced by increasing the neigh- borhood, i.e. by increasing the number of pairwise terms and adding proper weighting factors. In [Boy03, KB05] it was shown that anisotropies formulated as a certain class of metrics can asym- ptotically be approximated arbitrarily well using pairwise terms. However this requires the grid spa- cing and the neighborhood size to approach zero and infinity, respectively: For discretizations with a fixed number of neighbors, true isotropy cannot be guaranteed even for an arbitrarily fine grid. In practice, increasing the neighborhood size generally reduces artifacts but increases runtime. This effect is even more pronounced in higher- dimensional data [KSK+08]. 3.4. Other Approaches 3.4.1. Color TV For the restoration of multichannel data such as color images [Blo98] suggests to use .)(=)( 1/2 2 1=       i l i uTVuJ (59) While this approach is fully isotropic and per- mutation invariant, and seems to improve ROF restoration of images with large intensity devia- tion between color channels, it cannot be repre- sented in the form (22) using  . Moreover, it has the distinct disadvantage that it lacks locality, com- pletely coupling all points in the image. While this problem is less severe when using PDE-based schemes for optimization, it becomes a larger im- pediment when applying more advanced schemes that rely on some form of sparsity. 3.4.2. The lifting of nonconvex problems In [ABDM01, CCP08, PCBC09] a technique was developed to minimize general variational functionals of the form ,),,(:=)( dxuuxhuf   (60) over )(1,1  Wu , where f is convex in u , but not necessarily in u . The approach relies on the same approach as applied in [Ish03] in the graph cut framework, essentially lifting the problem originally formulated on dR to a higher- dimensional domain 1 dR . It was shown that  , {( , ) | ( ) } ( ) = 1 ,h x dx t u x t f u D         (61) where xh loc xh ,, :=   is defined implicitly via the Legendre-Fenchel conjugate of h with respect to the last argument:  )}.,,(|),(:=, vtxhwRRwv dxh loc   (62) Essentially, this transforms the problem of finding the optimal u into the problem of finding the set of points below its graph, which can be seen as a two-class segmentation problem in 1d . This can be solved by a relaxation technique ap- plied to (61). The problem can thus be treated as a highly anisotropic segmentation problem. 3.4.3. Partially separable norms For linearizations of labeling problems that in- volve a large number of labels at each point, op- timization can be made more efficient by exploit- ing separability in the regularizer. This occurs for example in optical flow estimation, where the two-dimensional flow vectors ),(= 21 uuu at each point are quantized using M2 labels, which re- quires a prohibitively large amount of memory for fine quantizations. If the regularizer decomposes with respect to 1u and 2u , i.e. )()(=)( 2211 uJuJuJ  , it is possi- ble to apply the relaxation technique in [GC10b], УСиМ, 2011, № 2 53 which only requires memory in the order of )(2MO as opposed to O(M2). 4. Conclusion In this paper, we tried to give an overview over recent variational methods that make use of the unique properties of total variation-based regular- izers. 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<FEFF04120438043a043e0440043804410442043e043204430439044204350020044604560020043f043004400430043c043504420440043800200434043b044f0020044104420432043e04400435043d043d044f00200434043e043a0443043c0435043d044204560432002000410064006f006200650020005000440046002c0020044f043a04560020043d04300439043a04400430044904350020043f045604340445043e0434044f0442044c00200434043b044f0020043204380441043e043a043e044f043a04560441043d043e0433043e0020043f0435044004350434043404400443043a043e0432043e0433043e0020043404400443043a0443002e00200020042104420432043e04400435043d045600200434043e043a0443043c0435043d0442043800200050004400460020043c043e0436043d04300020043204560434043a0440043804420438002004430020004100630072006f006200610074002004420430002000410064006f00620065002000520065006100640065007200200035002e0030002004300431043e0020043f04560437043d04560448043e04570020043204350440044104560457002e> /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. 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